A Better Butterfly Theorem

The following generalization of the Butterfly Problem has been pointed out to me by Qiu Fawen, a Chinese teacher who discovered the result with his students in 1997. The Chinese version was at some time available at qiusir.com. I had to do some guess work to figure out what it was about.

According to Leon Bankoff (Mathematics Magazine, Volume 60, No. 4, October 1987, pp. 195-210) the appellation The Butterfly made its first appearance in the Solutions section of the American Mathematical Monthly in the February 1944 issue. Since the diagram of the theorem (with two wings on each side) established by Qiu Fawen and his students gives a more realistic depiction of a butterfly, I suggest to call the statement A Better Butterfly Theorem which might be interpreted as both A "Better Butterfly" Theorem and A Better "Butterfly Theorem", the former being my preference.

A Better Butterfly Theorem

Let there be two concentric circles with the common center O. A line crosses the two circles at points P, Q and P', Q', M being the common midpoint of PQ and P'Q'. Through M, draw two lines AA'B'B and CC'D'D and connect AD', A'D, BC', and B'C. (This is the Butterfly.) Let X, Y, Z, W be the points of intersection of PP'Q'Q with AD', B'C, A'D, and BC', respectively. Then

(1) 1/MX + 1/MZ = 1/MY + 1/MW.

(Since X coincides with Z and Y with W when the two circles coalesce into one, The Butterfly Theorem is an immediate consequence of A Better Butterfly Theorem.)

The proof depends on the following


Let in ΔRST, RU be a cevian through vertex R. Introduce angles a = ∠SRU and b = ∠URT. Then

(2) sin(a + b)/RU = sin(a)/RT + sin(b)/RS.

The proof follows from the fact that Area( ΔRST) = Area( ΔRSU) + Area( ΔRUT) by a consistent application of the sine formula for the area of a triangle. (For example, Area( ΔRST) = RS·RT·sin(a + b)/2.)

Proof of the theorem

We apply Lemma to triangles AMD', A'MD, B'MC, and BMC':

(3) sin(a + b)/MX = sin(a)/MD' + sin(b)/MA,
sin(a + b)/MZ = sin(a)/MD + sin(b)/MA',
sin(a + b)/MY = sin(a)/MC + sin(b)/MB',
sin(a + b)/MW = sin(a)/MC' + sin(b)/MB.

In view of (3), (1) will follow from

(4) sin(a)/MD' + sin(b)/MA + sin(a)/MD + sin(b)/MA' =
      sin(a)/MC + sin(b)/MB' + sin(a)/MC' + sin(b)/MB,

or, which is the same, from

(5) sin(b)(1/MA - 1/MB) + sin(b)(1/MA' - 1/MB') =
      sin(a)(1/MC - 1/MD) + sin(a)(1/MC' - 1/MD').

Now, drop perpendiculars OM1 and OM2 from O onto AA'B'B and CC'D'D. M1 is the midpoint of both AB and A'B', whereas M2 is the midpoint of CD and C'D'. Obviously,

(6) MB - MA = MB' - MA' = 2·OM1 = 2·OM·sin(a) and
MD - MC = MD' - MC' = 2·OM2 = 2·OM·sin(b).

With (6) in mind, (5) is equivalent to

(7) 1/MA·MB + 1/MA'·MB' = 1/MC·MD + 1/MC'·MD'.

However, as is well known, MA·MB = MC·MD and MA'·MB' = MC'·MD'. Therefore, (7) is true, as is (5), which in turn, implies (1).

Butterfly Theorem and Variants

  1. Butterfly theorem
  2. 2N-Wing Butterfly Theorem
  3. Better Butterfly Theorem
  4. The Lepidoptera of the Circles
  5. The Lepidoptera of the Quadrilateral
  6. The Lepidoptera of the Quadrilateral II
  7. Butterflies in Ellipse
  8. Butterflies in Hyperbola
  9. Butterflies in Quadrilaterals and Elsewhere
  10. Pinning Butterfly on Radical Axes
  11. Shearing Butterflies in Quadrilaterals
  12. The Plain Butterfly Theorem
  13. Two Butterflies Theorem
  14. Two Butterflies Theorem II
  15. Two Butterflies Theorem III
  16. Algebraic proof of the theorem of butterflies in quadrilaterals
  17. William Wallace's Proof of the Butterfly Theorem
  18. Butterfly theorem, a Projective Proof
  19. Areal Butterflies
  20. Butterflies in Similar Co-axial Conics
  21. Butterfly Trigonometry
  22. Butterfly in Kite
  23. Butterfly with Menelaus
  24. William Wallace's 1803 Statement of the Butterfly Theorem
  25. Butterfly in Inscriptible Quadrilateral
  26. Camouflaged Butterfly
  27. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  28. Butterfly via Ceva
  29. Butterfly via the Scale Factor of the Wings
  30. Butterfly by Midline
  31. Stathis Koutras' Butterfly
  32. General Butterfly in Pictures
  33. Two Butterfly Theorems by Sidney Kung

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